The derivative at a point: the limit of the secants' slopes

Calculus, grade 10 (5 units) Lesson 9 of 26Learning

The derivative at a point: the limit of the secants' slopes

The tangent: the graph close up

When you zoom in on a graph around a point on it, it looks straighter and straighter, until the eye cannot tell it from a straight line. That line, the closest to the graph near the point, is the tangent to the graph at the point.

The board below shows the graph of $\displaystyle {f(x)}={x^2}$ and its tangent at the point $\displaystyle (1,1)$ . Zoom in around the point and watch the graph and the tangent come together.

\displaystyle f(x)=x^{2}

\displaystyle y=2x-1

\displaystyle A=(1,1)

The tangent describes where the graph goes near the point.
The slope of the tangent is positive – the function rises there; negative – it falls.
Here the slope of the tangent is 2. How is it worked out? That is what this lesson is about.

From a secant to the tangent

Take the point on the graph where $\displaystyle {x}={x_0}$ , and a point close to it where $\displaystyle {x}={x_0+h}$ . The slope of the secant through them is the average rate of change between them:

\frac{f(x_0+h)-f(x_0)}{h}

The smaller $\displaystyle h$ is, the closer the second point comes to the first, and the closer the secant comes to the tangent.
So the slope of the tangent is the number the slopes of the secants approach as $\displaystyle h$ approaches 0.
We write “ $\displaystyle h\to0$ ” and read “ $\displaystyle h$ tends to 0”. The number $\displaystyle h$ approaches is 0, but $\displaystyle h$ itself is never 0 – otherwise there is no secant, and no dividing by 0 either.

The speed of the alien car

An alien car travels along a track. After $\displaystyle x$ seconds it has covered $\displaystyle {f(x)}={2x^2}$ metres.

In the first 5 seconds the car covered $\displaystyle {f(5)}={50}$ metres, so its average speed was $\displaystyle {\frac{50}{5}}={10}$ metres per second. But what is its speed exactly at the moment $\displaystyle {x}={5}$ ?

The example below works out the average speed over shorter and shorter times after the fifth second:

In the thousandth of a second after $\displaystyle {x}={5}$ : $\displaystyle {\frac{f(5.001)-f(5)}{0.001}}={\frac{50.020002-50}{0.001}}={20.002}$ metres per second.
Over a general time $\displaystyle h$ :

\frac{f(5+h)-f(5)}{h}=\frac{2(5+h)^2-2\cdot5^2}{h}=\frac{20h+2h^2}{h}=20+2h

As $\displaystyle h\to0$ , the expression $\displaystyle 20+2h$ tends to 20.

The car’s speed at the moment $\displaystyle {x}={5}$ is 20 metres per second: this is the instantaneous speed, what a car’s speedometer shows.

The derivative at a point

The derivative of $\displaystyle f$ at the point $\displaystyle x_0$ , written $\displaystyle f'(x_0)$ , is the number the slopes of the secants tend to as $\displaystyle h\to0$ . With the limit sign (lim):

\displaystyle f'(x_0)

\displaystyle =

\displaystyle \lim_{h\to0}\frac{f(x_0+h)-f(x_0)}{h}

$\displaystyle f'(x_0)$ is the slope of the tangent to the graph at the point where $\displaystyle {x}={x_0}$ .
$\displaystyle f'(x_0)$ is the instantaneous rate of change of $\displaystyle f$ at $\displaystyle x_0$ ; when $\displaystyle f$ is distance against time – the instantaneous speed.
It can also be written with arrows: $\displaystyle \frac{f(x_0+h)-f(x_0)}{h}\xrightarrow[h\to0]{}f'(x_0)$ .

Example:

\displaystyle {f(x)}={x^2-3x}

\displaystyle {x_0}={2}

The example below finds $\displaystyle f'(2)$ for $\displaystyle {f(x)}={x^2-3x}$ by the definition.

$\displaystyle {f(2)}={4-6}={-2}$ .
$\displaystyle {f(2+h)}={(2+h)^2-3(2+h)}={4+4h+h^2-6-3h}={h^2+h-2}$ .
The slope of the secant: $\displaystyle {\frac{f(2+h)-f(2)}{h}}={\frac{h^2+h}{h}}={h+1}$ .
As $\displaystyle h\to0$ , $\displaystyle h+1$ tends to 1, so $\displaystyle {f'(2)}={1}$ .

Note the order: first simplify the fraction and cancel $\displaystyle h$ , and only then ask what the expression tends to. Putting $\displaystyle {h}={0}$ into the fraction itself gives $\displaystyle \frac{0}{0}$ , which is not a number.

Exercise 1

The function is $\displaystyle {f(x)}={x^2}$ .

Simplify the slope of the secant between $\displaystyle {x}={3}$ and $\displaystyle {x}={3+h}$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

What does your expression tend to as $\displaystyle h\to0$ ?
Write $\displaystyle f'(3)$ .

$\displaystyle f(x)=x^{2}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

The function is $\displaystyle {f(x)}={x^2-4x+1}$ .
$\displaystyle {f(1)}={-2}$ .

Simplify the slope of the secant between $\displaystyle {x}={1}$ and $\displaystyle {x}={1+h}$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Write $\displaystyle f'(1)$ .

$\displaystyle f(x)=x^{2}-4x+1$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The board below shows the graph of $\displaystyle f$ and its tangent at the point $\displaystyle (1,-2)$ .
Does the slope of the tangent match your answer?
Does the function rise or fall near $\displaystyle {x}={1}$ ?

\displaystyle f(x)=x^{2}-4x+1

\displaystyle y=-2x

\displaystyle A=(1,-2)

Exercise 3

The function is $\displaystyle {f(x)}={x^3}$ .

Simplify the slope of the secant between $\displaystyle {x}={2}$ and $\displaystyle {x}={2+h}$ .
Use $\displaystyle {(2+h)^3}={8+12h+6h^2+h^3}$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Write $\displaystyle f'(2)$ .

$\displaystyle f(x)=x^{3}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

A ball is thrown straight up.
After $\displaystyle t$ seconds its height is $\displaystyle {s(t)}={20t-5t^2}$ metres.

Simplify the ball’s average speed between $\displaystyle {t}={1}$ and $\displaystyle {t}={1+h}$ .

$\displaystyle s(t)=20t-5t^{2}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

What is the ball’s instantaneous speed after one second?

$\displaystyle s(t)=20t-5t^{2}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

What is the ball’s instantaneous speed after 2 seconds?

$\displaystyle s(t)=20t-5t^{2}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

What happens to the ball at the moment $\displaystyle {t}={2}$ ?
Explain by your answer to part c.

Exercise 5

The linear function is $\displaystyle {f(x)}={5-2x}$ .

Simplify the slope of the secant between $\displaystyle {x}={3}$ and $\displaystyle {x}={3+h}$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

What is the derivative of $\displaystyle f$ at $\displaystyle {x}={3}$ ?
Explain why the derivative of a linear function equals its slope at every point.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: calculus, grade 10 (5 units)

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Algebra

The square of a sum and of a difference
$\displaystyle (a\pm b)^2$ $\displaystyle =$ $\displaystyle a^2\pm2ab+b^2$
The difference of two squares
$\displaystyle a^2-b^2$ $\displaystyle =$ $\displaystyle (a-b)(a+b)$
The cube of a sum and of a difference
$\displaystyle (a\pm b)^3$ $\displaystyle =$ $\displaystyle a^3\pm3a^2b+3ab^2\pm b^3$
The sum and the difference of two cubes
$\displaystyle a^3\pm b^3$ $\displaystyle =$ $\displaystyle (a\pm b)(a^2\mp ab+b^2)$
The quadratic formula for $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Factoring a trinomial – $\displaystyle x_1$ , $\displaystyle x_2$ the roots of $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle ax^2+bx+c$ $\displaystyle =$ $\displaystyle a(x-x_1)(x-x_2)$

The properties of a function

An even function – the graph is symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – the graph is symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$
The root of a square
$\displaystyle \sqrt{x^2}$ $\displaystyle =$ $\displaystyle |x|$
The absolute value
$\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}$

Transformations of a graph

A shift of $\displaystyle k$ units up ( $\displaystyle k<0$ – down)
$\displaystyle y=f(x)+k$
A shift of $\displaystyle p$ units right ( $\displaystyle p<0$ – left)
$\displaystyle y=f(x-p)$
A vertical stretch ( $\displaystyle |a|>1$ ) or squeeze ( $\displaystyle |a|<1$ )
$\displaystyle y=a\cdot f(x)$
A horizontal squeeze by $\displaystyle a$ ( $\displaystyle a>1$ ) or stretch ( $\displaystyle 0<a<1$ )
$\displaystyle y=f(ax)$
A reflection in the $\displaystyle x$ -axis
$\displaystyle y=-f(x)$
A reflection in the $\displaystyle y$ -axis
$\displaystyle y=f(-x)$

The derivative

The average rate of change from $\displaystyle x_1$ to $\displaystyle x_2$ – the slope of the secant
$\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}$
The derivative at a point – the slope of the tangent
$\displaystyle f'(x_0)=\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}{h}$
The derivative of a power
$\displaystyle (x^n)'$ $\displaystyle =$ $\displaystyle nx^{n-1}$
The derivative of a constant
$\displaystyle (c)'$ $\displaystyle =$ $\displaystyle 0$
The derivative of a constant multiple
$\displaystyle (c\cdot f(x))'$ $\displaystyle =$ $\displaystyle c\cdot f'(x)$
The derivative of a sum and of a difference
$\displaystyle (f(x)\pm g(x))'$ $\displaystyle =$ $\displaystyle f'(x)\pm g'(x)$
The instantaneous speed – the derivative of the distance with respect to time
$\displaystyle v(t)$ $\displaystyle =$ $\displaystyle s'(t)$

Rules of differentiation

The derivative of a product of functions
$\displaystyle [f(x)\cdot g(x)]'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a quotient of functions
$\displaystyle \left[\frac{f(x)}{g(x)}\right]'$ $\displaystyle =$ $\displaystyle \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^2}$
The derivative of a composite function – the outer derivative times the inner derivative
$\displaystyle [f(u(x))]'$ $\displaystyle =$ $\displaystyle f'(u)\cdot u'(x)$
The derivative of $\displaystyle \frac{1}{x}$
$\displaystyle \left(\frac{1}{x}\right)'$ $\displaystyle =$ $\displaystyle -\frac{1}{x^2}$
The derivative of $\displaystyle \sqrt{x}$
$\displaystyle (\sqrt{x})'$ $\displaystyle =$ $\displaystyle \frac{1}{2\sqrt{x}}$ $\displaystyle x>0$
The derivative of the root of a function
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

The tangent to a graph

The slope of the tangent at the point $\displaystyle (x_1,y_1)$ on the graph
$\displaystyle m=f'(x_1)$
The equation of the tangent
$\displaystyle y-y_1=m(x-x_1)$

Rising, falling and extremum points

The function rises
$\displaystyle f'(x)>0$
The function falls
$\displaystyle f'(x)<0$
A point that may be an extremum
$\displaystyle f'(x)=0$

Asymptotes

A vertical asymptote – $\displaystyle a$ a zero of the denominator that is not a zero of the numerator
$\displaystyle x=a$
A horizontal asymptote – $\displaystyle b$ the value the function approaches as $\displaystyle x$ grows or falls without end
$\displaystyle y=b$